Can you win 3-card monte?

You walk up to a busy city corner and spot a lone street-magician with three cards on a table. Interested, you ask him what magic could possibly come out of these cards. He flips over one of the cards and shows you an ace of spades. He asks you for $5. He proposes to you that if you can keep track of this ace of spades (A♠) he’ll give you $10 back.

So, you either lose $5 or win $5. Thus, to win the game, you would need to pick the right card with a probability of 50% or more for a positive expected value of your winnings. Confident in your abilities to track his hands, you believe that will help you have a higher probability of picking the right card. So, you agree to the terms and in an instant he flips the cards over and shuffles the three cards multiple times, and stops, leaving you to select which card is the queen.

So here’s the question:
How relevant is your skill in winning tracking the cards as the number of shuffles the magician makes increases just a little?

Game Setup

In a typical setup, three cards are laid out on a table, one of which is the “money card”, in this case A♠, that if kept track of rewards a prize.

Selecting one of the other two cards means you lose your bet.

Let’s say the two loser cards are 2♠, 3♠, and as previously mentioned, the “money card” is the A♠.

So the game setup looks like this:

Once you see the starting position of the cards, they are flipped over to conceal their faces. The magician proceeds to shuffle, or swap the position of any two of the three cards, and it is the player’s job to select the A♠ when he finishes.

Theory

Uncertainity in this game or at least part of the uncertainty comes from not registering a shuffle in your brain. So, skill in this game can be represented by an accuracy $\alpha$ that you successfully track a shuffle of two cards. That is,

the probability that you know the exact position of the winner card after the shuffle takes place.

Suppose we know that the cards initially are (2♠, A♠, 3♠). Written in a tuple, the probabilities of the left, middle, and right cards being the winner, A♠, are (0,1,0), with the 1 in the middle representing 100% confidence that the winning card is there. Now, suppose you believe a shuffle happened between the middle and right cards with accuracy $\alpha$ . The new probabilities would be $(0,(1-\alpha) \cdot 1+\alpha \cdot 0, \alpha \cdot 1 + (1-\alpha) \cdot 0)$

In general, if the initial probabilities were : $(p_0, p_1, p_2)$

And you believed a shuffle happened between the middle and right cards, the new probabilities would be: $(p_0, (1-\alpha) \cdot p_1+\alpha \cdot p_2, (1-\alpha) \cdot p_2+\alpha \cdot p_1)$

Let’s play

Imagine you had decent accuracy at tracking the shuffles. Say your accuracy is 0.6. Try flipping the cards and shuffling them

Remember, when all probabilities drop under 50% marked by the red dashed line, you’re expected to lose money!

So, you see that it only takes a few shuffles after which the probability of all cards go below 0.5. At that point, you lose your advantage and the magician is likely to make money off you.

Let’s play with a higher accuracy

Suppose you were indeed a person of great skill, possessing sharper eyes than most. Your accuracy would be around 0.8. Would that make a difference?

Remember, when all probabilities drop under 50% marked by the red dashed line, you’re expected to lose money!

You see that regardless of your accuracy it only takes a few more shuffles until the magician takes the advantage from you.

You can play more below by changing the accuracy below, restarting and trying new simulations.

Your turn to experiment!

Accuracy: Restart

Remember, when all probabilities drop under 50% marked by the red dashed line, you’re expected to lose money!

Have fun playing!

Acknowledgements

• Idyll is the tool used to create interactive stories like these.
• The layout of the article was inspired by and adapted from the article on card shuffling by Fred Hohman “The math of shuffling cards”. Some parts of the code for the visualization of the cards has been adapted from the code written by Fred Hohman.
• Our code can be found here.
• The cover image was obtained from Creative Commons. It has the following attribution: “SHELL GAME” by Julia Boden is licensed under CC BY-NC-ND 4.0.